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Summary

This third edition provides a simple, basic approach to the finite element method that can be understood by readers. It does not have the usual prerequisites (such as structural analysis) required by most available books in this area. The book is written primarily as a basic learning tool for civil and mechanical engineering readers whose main interest is in stress analysis and heat transfer. The book is geared toward those who want to apply the finite element method as a tool to solve practical physical problems.

Table of Contents

Introduction

1

(25)

Prologue

1

(1)

Brief History

2

(1)

Introduction to Matrix Notation

3

(3)

Role of the Computer

6

(1)

General Steps of the Finite Element Method

6

(7)

Applications of the Finite Element Method

13

(5)

Advantages of the Finite Element Method

18

(1)

Computer Programs for the Finite Element Method

19

(7)

References

22

(3)

Problems

25

(1)

Introduction to the Stiffness (Displacement) Method

26

(37)

Introduction

26

(1)

Definition of the Stiffness Matrix

26

(1)

Derivation of the Stiffness Matrix for a Spring Element

27

(5)

Example of a Spring Assemblage

32

(3)

Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)

35

(2)

Boundary Conditions

37

(13)

Potential Energy Approach to Derive Spring Element Equations

50

(13)

References

58

(1)

Problems

59

(4)

Development of Truss Equations

63

(74)

Introduction

63

(1)

Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates

63

(6)

Selecting Approximation Functions for Displacements

69

(2)

Transformation of Vectors in Two Dimensions

71

(3)

Global Stiffness Matrix

74

(4)

Computation of Stress for a Bar in the x-y Plane

78

(2)

Solution of a Plane Truss

80

(7)

Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space

87

(5)

Use of Symmetry in Structure

92

(3)

Inclined, or Skewed, Supports

95

(6)

Potential Energy Approach to Derive Bar Element Equations

101

(11)

Comparison of Finite Element Solution to Exact Solution for Bar

112

(4)

Galerkin's Residual Method and Its Application to a One-Dimensional Bar

116

(21)

References

119

(1)

Problems

120

(17)

Development of Beam Equations

137

(51)

Introduction

137

(1)

Beam Stiffness

138

(5)

Example of Assemblage of Beam Stiffness Matrices

143

(2)

Examples of Beam Analysis Using the Direct Stiffness Method

145

(9)

Distributed Loading

154

(11)

Comparision of the Finite Element Solution to the Exact Solution for a Beam

165

(6)

Beam Element with Nodal Hinge

171

(5)

Potential Energy Approach to Derive Beam Element Equations

176

(3)

Galerkin's Method for Deriving Beam Element Equations

179

(9)

References

181

(1)

Problems

181

(7)

Frame and Grid Equations

188

(76)

Introduction

188

(1)

Two-Dimensional Arbitrarily Oriented Beam Element

188

(4)

Rigid Plane Frame Examples

192

(19)

Inclined or Skewed Supports-Frame Element

211

(1)

Grid Equations

212

(17)

Beam Element Arbitrarily Oriented in Space

229

(5)

Concept of Substructure Analysis

234

(30)

References

240

(1)

Problems

240

(24)

Development of the Plane Stress and Plane Strain Stiffness Equations

264

(43)

Introduction

264

(1)

Basic Concepts of Plane Stress and Plane Strain

265

(5)

Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations